Calcul numérique de la transformée de Fourier de fonctions usuelles avec python

Quelques exemples de comment calculer numériquement une transformée de Fourier pour quelques fonctions usuelles avec python en utilisant scipy fft

Fonction cosinus

import numpy as np
import matplotlib.pyplot as plt
import scipy.fftpack

#----------------------------------------------------------------------------------------#

N = 50000 # Number of samplepoints

T = 1.0 / 1000.0 # sample spacing

x = np.linspace(0.0, N*T, N)

y = np.cos(2.0*np.pi*x)

plt.plot(x,y)

plt.xlim(0,3.0*np.pi)

plt.title(r'$cos(2\pi \nu x)$ with $\nu=1$')

plt.savefig('fourrier_transform_cosinus.png', bbox_inches='tight')

plt.close()

yf = scipy.fftpack.fft(y)

yf = np.fft.fftshift(yf)

xf = np.linspace(-1.0/(2.0*T), 1.0/(2.0*T), N)

fig, ax = plt.subplots()

ax.plot(xf, 1.0/N *np.abs(yf) )

plt.xlim(-4,4)
plt.title('FFT (cosinus function) power spectrum')

plt.grid()

plt.savefig('fourrier_transform_cosinus_abs.png', bbox_inches='tight')

plt.close()

Fonction sinus

N = 50000 # Number of samplepoints

T = 1.0 / 1000.0 # sample spacing

x = np.linspace(0.0, N*T, N)

y = np.sin(2.0*np.pi*x)

plt.plot(x,y)

plt.xlim(0,3.0*np.pi)

plt.title(r'$sin(2\pi \nu x)$ with $\nu=1$')

plt.savefig('fourrier_transform_sinus.png', bbox_inches='tight')

plt.close()

yf = scipy.fftpack.fft(y)

yf = np.fft.fftshift(yf)

xf = np.linspace(-1.0/(2.0*T), 1.0/(2.0*T), N)

fig, ax = plt.subplots()

ax.plot(xf, 1.0/N * np.abs(yf) )

plt.xlim(-4,4)
plt.title('FFT (sinus function) power spectrum')

plt.grid()

plt.savefig('fourrier_transform_sinus_abs.png', bbox_inches='tight')

plt.close()

Fonction rectangulaire

N = 50000 # Number of samplepoints

T = 1.0 / 1000.0 # sample spacing

x = np.linspace(0.0, N*T, N)

y = np.zeros(x.shape)

for i in range(x.shape[0]):
    if x[i] > -0.5 and x[i] < 0.5: 
        y[i] = 1.0

plt.plot(x,y)

plt.xlim(-2,2)

plt.title(r'Rectangular function')

plt.savefig('fourrier_transform_rectangular.png', bbox_inches='tight')

plt.close()

yf = scipy.fftpack.fft(y)

yf = np.fft.fftshift(yf)

xf = np.linspace(-1.0/(2.0*T), 1.0/(2.0*T), N)

fig, ax = plt.subplots()

ax.plot(xf, np.abs(yf) )

plt.xlim(-10,10)
plt.title('FFT (rectangular function) power spectrum')

plt.grid()

plt.savefig('fourrier_transform_rectangular_abs.png', bbox_inches='tight')

plt.close()

fig, ax = plt.subplots()

ax.plot(xf, np.real(yf) )

plt.xlim(-10,10)
plt.title('FFT (rectangular function) real')

plt.grid()

plt.savefig('fourrier_transform_rectangular_real.png', bbox_inches='tight')

plt.close()

fig, ax = plt.subplots()

ax.plot(xf, np.imag(yf) )

plt.xlim(-10,10)
plt.title('FFT (rectangular function) img')

plt.grid()

plt.savefig('fourrier_transform_rectangular_imag.png', bbox_inches='tight')

plt.close()

Références